What is the difference between what would you have me do? and what do you want me to do and what would you have me TO do ?Feel free to just provide example sentences.

"What would you have me do?" is for when you say you are going to do something and someone objects. Ex. "I am going to tell the truth!" "Are you trying to get grounded!?" "Well, what would you have me do?" "Lie!"

"What would you have me to do?" can be the same as without the "to". Or it could be used when talking about assigning people tasks. Ex. "We are all ready for the bake sale!" "Good. What are you doing for the sale?" "I am bringing brownies." "Oh, but Mrs.Johnson makes the best brownies!" "Well what would you have me to do?" But even this example could still leave off the "to" and sound natural.

"What do you want me to do?" Is simply asking for instruction.